The general behavior of $NLO$ unintegrated parton distributions based on the single-scale evolution and the angular ordering constraint

To overcome the complexity of generalized two hard scale ($k_t$,$\mu$) evolution equation, well known as the $Ciafaloni$, $Catani$, $Fiorani$ and $Marchsini$ ($CCFM$) evolution equations, and calculate the unintegrated parton distribution functions ($UPDF$), $Kimber$, $Martin$ and $Ryskin$ ($KMR$) proposed a procedure based on ($i$) the inclusion of single-scale ($\mu$) only at the last step of evolution and ($ii$) the angular ordering constraint ($AOC$) on the $DGLAP$ terms (the $DGLAP$ collinear approximation), to bring the second scale, $k_t$ into the $UPDF$ evolution equations. In this work we intend to use the $MSTW 2008$ (Martin et al) parton distribution functions (PDF) and try to calculate $UPDF$ for various values of $x$ (the longitudinal fraction of parton momentum), $\mu$ (the probe scale) and $k_t$ (the parton transverse momentum) to see the general behavior of three dimensional $UPDF$ at the $NLO$ level up to the $LHC$ working energy scales ($\mu^2)$. It is shown that there exits some pronounced peaks for the three dimensional $UPDF$ $(f_a(x,k_t))$ with respect to the two variables $x$ and $k_t$ at various energies ($\mu$). These peaks get larger and move to larger values of $k_t$, as the energy ($\mu$) is increased. We hope these peaks could be detected in the $LHC$ experiments at $CERN$ and other laboratories in the less exclusive processes.